Variational Bridge Regression

Here we obtain approximate Bayes inferences through variational methods when an exponential power family type prior is specified for the regression coefficients to mimic the characteristics of the Bridge regression. We accomplish this through hierarchical modeling of such priors. Although the mixing distribution is not explicitly stated for scale normal mixtures, we obtain the required moments only to attain the variational distributions for the regression coefficients. By choosing specific values of hyper-parameters (tuning parameters) present in the model, we can mimic the model selection performance of best subset selection in sparse underlying settings. The fundamental difference between MAP, maximum a posteriori, estimation and the proposed method is that, here we can obtain approximate inferences besides a point estimator. We also empirically analyze the frequentist properties of the estimator obtained. Results suggest that the proposed method yields an estimator that performs significantly better in sparse underlying setups than the existing state-of-the-art procedures in both n > p and p > n scenarios.